Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 85.2%

Time: 22.4s

Alternatives: 20

Speedup: 3.8×

• 27.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \sqrt[3]{\sin k}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{+43}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + t_1\right)} \cdot \left(t \cdot t_2\right)\right)}^{3}}\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t_2 \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)) (t_2 (cbrt (sin k))))
   (if (<= t -8.8e+43)
     (* l (* l (/ 2.0 (pow (* (cbrt (* (tan k) (+ 2.0 t_1))) (* t t_2)) 3.0))))
     (if (<= t 8e-49)
       (/ 2.0 (* (* (/ (* t k) l) (/ k l)) (* (tan k) (sin k))))
       (/
        2.0
        (*
         (pow (* t_2 (* t (cbrt (pow l -2.0)))) 3.0)
         (* (tan k) (+ 1.0 (+ t_1 1.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = cbrt(sin(k));
	double tmp;
	if (t <= -8.8e+43) {
		tmp = l * (l * (2.0 / pow((cbrt((tan(k) * (2.0 + t_1))) * (t * t_2)), 3.0)));
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	} else {
		tmp = 2.0 / (pow((t_2 * (t * cbrt(pow(l, -2.0)))), 3.0) * (tan(k) * (1.0 + (t_1 + 1.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = Math.cbrt(Math.sin(k));
	double tmp;
	if (t <= -8.8e+43) {
		tmp = l * (l * (2.0 / Math.pow((Math.cbrt((Math.tan(k) * (2.0 + t_1))) * (t * t_2)), 3.0)));
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = 2.0 / (Math.pow((t_2 * (t * Math.cbrt(Math.pow(l, -2.0)))), 3.0) * (Math.tan(k) * (1.0 + (t_1 + 1.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = cbrt(sin(k))
	tmp = 0.0
	if (t <= -8.8e+43)
		tmp = Float64(l * Float64(l * Float64(2.0 / (Float64(cbrt(Float64(tan(k) * Float64(2.0 + t_1))) * Float64(t * t_2)) ^ 3.0))));
	elseif (t <= 8e-49)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_2 * Float64(t * cbrt((l ^ -2.0)))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(t_1 + 1.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t, -8.8e+43], N[(l * N[(l * N[(2.0 / N[Power[N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-49], N[(2.0 / N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$2 * N[(t * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.80000000000000002e43

   1. Initial program 58.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 58.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 58.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 59.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 59.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 59.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 59.6%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 80.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. pow1 80.8%

         \[\leadsto \color{blue}{{\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)}^{1}} \]

      2. associate-*l* 83.9%

         \[\leadsto {\color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}}^{1} \]

   7. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}^{1}} \]

   if -8.80000000000000002e43 < t < 7.99999999999999949e-49

   1. Initial program 46.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 45.3%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 45.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 45.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 45.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 51.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 51.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 51.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 51.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 51.5%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 71.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 81.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 81.8%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 81.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around 0 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   10. Step-by-step derivation

       1. *-commutative 71.6%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

       2. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

       3. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       4. associate-*r* 75.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       5. times-frac 94.3%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Simplified 94.3%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 7.99999999999999949e-49 < t

   1. Initial program 76.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 76.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. pow3 76.6%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. cbrt-prod 76.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\sin k}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. div-inv 76.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. cbrt-prod 77.7%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. unpow3 77.7%

         \[\leadsto \frac{2}{{\left(\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. add-cbrt-cube 84.3%

         \[\leadsto \frac{2}{{\left(\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. pow2 84.3%

         \[\leadsto \frac{2}{{\left(\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      9. pow-flip 84.3%

         \[\leadsto \frac{2}{{\left(\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      10. metadata-eval 84.3%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. Applied egg-rr 84.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+43}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]

Alternative 2: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(1 + \left(t_1 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 4 \cdot 10^{+218}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\frac{\left(\tan k \cdot \left(2 + t_1\right)\right) \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ t_1 1.0))
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        4e+218)
     (*
      l
      (/ l (/ (* (* (tan k) (+ 2.0 t_1)) (pow (* t (cbrt (sin k))) 3.0)) 2.0)))
     (/ 2.0 (* (* (/ (* t k) l) (/ k l)) (* (tan k) (sin k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 4e+218) {
		tmp = l * (l / (((tan(k) * (2.0 + t_1)) * pow((t * cbrt(sin(k))), 3.0)) / 2.0));
	} else {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 4e+218) {
		tmp = l * (l / (((Math.tan(k) * (2.0 + t_1)) * Math.pow((t * Math.cbrt(Math.sin(k))), 3.0)) / 2.0));
	} else {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(t_1 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 4e+218)
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(tan(k) * Float64(2.0 + t_1)) * (Float64(t * cbrt(sin(k))) ^ 3.0)) / 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+218], N[(l * N[(l / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 4.00000000000000033e218

   1. Initial program 82.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 82.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 82.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 80.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 80.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 80.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 80.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 81.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 81.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 81.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 81.5%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 81.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 81.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 86.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. pow1 86.3%

         \[\leadsto \color{blue}{{\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)}^{1}} \]

      2. associate-*l* 89.2%

         \[\leadsto {\color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}}^{1} \]

   7. Applied egg-rr 89.2%

      \[\leadsto \color{blue}{{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. associate-*r/ 89.9%

         \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}}\right)}^{1} \]

   9. Applied egg-rr 89.9%

      \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell \cdot 2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}}\right)}^{1} \]
   10. Step-by-step derivation

       1. associate-/l* 89.9%

          \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell}{\frac{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}{2}}}\right)}^{1} \]

       2. *-commutative 89.9%

          \[\leadsto {\left(\ell \cdot \frac{\ell}{\frac{{\color{blue}{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}}^{3}}{2}}\right)}^{1} \]

       3. cube-prod 88.0%

          \[\leadsto {\left(\ell \cdot \frac{\ell}{\frac{\color{blue}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot {\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}}{2}}\right)}^{1} \]

       4. rem-cube-cbrt 88.0%

          \[\leadsto {\left(\ell \cdot \frac{\ell}{\frac{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{2}}\right)}^{1} \]

       5. *-commutative 88.0%

          \[\leadsto {\left(\ell \cdot \frac{\ell}{\frac{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}}{2}}\right)}^{1} \]

   11. Simplified 88.0%

       \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell}{\frac{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}{2}}}\right)}^{1} \]

   if 4.00000000000000033e218 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 24.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 24.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 23.3%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 23.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 23.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 23.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 23.3%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 23.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 23.3%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 23.3%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 22.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 22.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 22.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 37.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 37.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 37.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 37.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 37.7%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 59.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 59.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 59.8%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 68.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 68.3%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 68.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around 0 59.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   10. Step-by-step derivation

       1. *-commutative 59.8%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

       2. unpow2 59.8%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

       3. unpow2 59.8%

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       4. associate-*r* 63.8%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       5. times-frac 83.7%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Simplified 83.7%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 4 \cdot 10^{+218}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \]

Alternative 3: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\\ \mathbf{if}\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 10^{+244}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot t_1\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ (pow (/ k t) 2.0) 1.0))))
   (if (<=
        (/ 2.0 (* t_1 (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        1e+244)
     (/ 2.0 (* (* (tan k) t_1) (* (/ (pow t 3.0) l) (/ (sin k) l))))
     (/ 2.0 (* (* (/ (* t k) l) (/ k l)) (* (tan k) (sin k)))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (pow((k / t), 2.0) + 1.0);
	double tmp;
	if ((2.0 / (t_1 * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 1e+244) {
		tmp = 2.0 / ((tan(k) * t_1) * ((pow(t, 3.0) / l) * (sin(k) / l)));
	} else {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (((k / t) ** 2.0d0) + 1.0d0)
    if ((2.0d0 / (t_1 * (tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))))) <= 1d+244) then
        tmp = 2.0d0 / ((tan(k) * t_1) * (((t ** 3.0d0) / l) * (sin(k) / l)))
    else
        tmp = 2.0d0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (Math.pow((k / t), 2.0) + 1.0);
	double tmp;
	if ((2.0 / (t_1 * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 1e+244) {
		tmp = 2.0 / ((Math.tan(k) * t_1) * ((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)));
	} else {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 1.0 + (math.pow((k / t), 2.0) + 1.0)
	tmp = 0
	if (2.0 / (t_1 * (math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))))) <= 1e+244:
		tmp = 2.0 / ((math.tan(k) * t_1) * ((math.pow(t, 3.0) / l) * (math.sin(k) / l)))
	else:
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (math.tan(k) * math.sin(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0))
	tmp = 0.0
	if (Float64(2.0 / Float64(t_1 * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 1e+244)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * t_1) * Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 1.0 + (((k / t) ^ 2.0) + 1.0);
	tmp = 0.0;
	if ((2.0 / (t_1 * (tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))))) <= 1e+244)
		tmp = 2.0 / ((tan(k) * t_1) * (((t ^ 3.0) / l) * (sin(k) / l)));
	else
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+244], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 1.00000000000000007e244

   1. Initial program 82.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 82.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 82.5%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 82.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in t around 0 83.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 69.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot \sin k}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)} \]

      2. unpow2 69.7%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

      3. times-frac 70.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)} \cdot \left(2 \cdot k\right)} \]

   6. Simplified 86.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if 1.00000000000000007e244 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 23.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 23.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 23.5%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 23.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 23.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 23.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 23.5%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 23.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 23.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 23.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 22.6%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 22.6%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 22.6%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 37.8%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 37.8%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 38.0%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 38.0%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 38.0%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 60.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 60.3%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 60.3%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 68.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 68.9%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 68.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around 0 60.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   10. Step-by-step derivation

       1. *-commutative 60.3%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

       2. unpow2 60.3%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

       3. unpow2 60.3%

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       4. associate-*r* 64.3%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       5. times-frac 84.4%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Simplified 84.4%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 10^{+244}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \]

Alternative 4: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(1 + \left(t_1 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 4 \cdot 10^{+218}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + t_1\right) \cdot \left(\sin k \cdot {t}^{3}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ t_1 1.0))
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        4e+218)
     (* l (* l (/ 2.0 (* (tan k) (* (+ 2.0 t_1) (* (sin k) (pow t 3.0)))))))
     (/ 2.0 (* (* (/ (* t k) l) (/ k l)) (* (tan k) (sin k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 4e+218) {
		tmp = l * (l * (2.0 / (tan(k) * ((2.0 + t_1) * (sin(k) * pow(t, 3.0))))));
	} else {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if ((2.0d0 / ((1.0d0 + (t_1 + 1.0d0)) * (tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))))) <= 4d+218) then
        tmp = l * (l * (2.0d0 / (tan(k) * ((2.0d0 + t_1) * (sin(k) * (t ** 3.0d0))))))
    else
        tmp = 2.0d0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 4e+218) {
		tmp = l * (l * (2.0 / (Math.tan(k) * ((2.0 + t_1) * (Math.sin(k) * Math.pow(t, 3.0))))));
	} else {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if (2.0 / ((1.0 + (t_1 + 1.0)) * (math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))))) <= 4e+218:
		tmp = l * (l * (2.0 / (math.tan(k) * ((2.0 + t_1) * (math.sin(k) * math.pow(t, 3.0))))))
	else:
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (math.tan(k) * math.sin(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(t_1 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 4e+218)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(Float64(2.0 + t_1) * Float64(sin(k) * (t ^ 3.0)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))))) <= 4e+218)
		tmp = l * (l * (2.0 / (tan(k) * ((2.0 + t_1) * (sin(k) * (t ^ 3.0))))));
	else
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+218], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + t$95$1), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 4.00000000000000033e218

   1. Initial program 82.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 82.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 82.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 80.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 80.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 80.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 80.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 81.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 81.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 81.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 81.5%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 66.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 60.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

   5. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 66.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-log1p 81.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

      3. associate-*l* 84.1%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} \]

      4. *-commutative 84.1%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sin k \cdot {t}^{3}\right)}\right)}\right) \]

   7. Simplified 84.1%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {t}^{3}\right)\right)}\right)} \]

   if 4.00000000000000033e218 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 24.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 24.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 23.3%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 23.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 23.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 23.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 23.3%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 23.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 23.3%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 23.3%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 22.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 22.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 22.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 37.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 37.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 37.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 37.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 37.7%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 59.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 59.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 59.8%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 68.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 68.3%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 68.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around 0 59.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   10. Step-by-step derivation

       1. *-commutative 59.8%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

       2. unpow2 59.8%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

       3. unpow2 59.8%

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       4. associate-*r* 63.8%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       5. times-frac 83.7%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Simplified 83.7%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 4 \cdot 10^{+218}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {t}^{3}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \]

Alternative 5: 83.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \sqrt[3]{\sin k}\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + t_1\right)} \cdot \left(t \cdot t_2\right)\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t_2 \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)) (t_2 (cbrt (sin k))))
   (if (<= t -4.4e+44)
     (* (/ 2.0 (pow (* (cbrt (* (tan k) (+ 2.0 t_1))) (* t t_2)) 3.0)) (* l l))
     (if (<= t 8e-49)
       (/ 2.0 (* (* (/ (* t k) l) (/ k l)) (* (tan k) (sin k))))
       (/
        2.0
        (*
         (pow (* t_2 (* t (cbrt (pow l -2.0)))) 3.0)
         (* (tan k) (+ 1.0 (+ t_1 1.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = cbrt(sin(k));
	double tmp;
	if (t <= -4.4e+44) {
		tmp = (2.0 / pow((cbrt((tan(k) * (2.0 + t_1))) * (t * t_2)), 3.0)) * (l * l);
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	} else {
		tmp = 2.0 / (pow((t_2 * (t * cbrt(pow(l, -2.0)))), 3.0) * (tan(k) * (1.0 + (t_1 + 1.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = Math.cbrt(Math.sin(k));
	double tmp;
	if (t <= -4.4e+44) {
		tmp = (2.0 / Math.pow((Math.cbrt((Math.tan(k) * (2.0 + t_1))) * (t * t_2)), 3.0)) * (l * l);
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = 2.0 / (Math.pow((t_2 * (t * Math.cbrt(Math.pow(l, -2.0)))), 3.0) * (Math.tan(k) * (1.0 + (t_1 + 1.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = cbrt(sin(k))
	tmp = 0.0
	if (t <= -4.4e+44)
		tmp = Float64(Float64(2.0 / (Float64(cbrt(Float64(tan(k) * Float64(2.0 + t_1))) * Float64(t * t_2)) ^ 3.0)) * Float64(l * l));
	elseif (t <= 8e-49)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_2 * Float64(t * cbrt((l ^ -2.0)))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(t_1 + 1.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t, -4.4e+44], N[(N[(2.0 / N[Power[N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-49], N[(2.0 / N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$2 * N[(t * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.39999999999999991e44

   1. Initial program 58.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 58.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 58.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 59.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 59.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 59.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 59.6%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 80.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   if -4.39999999999999991e44 < t < 7.99999999999999949e-49

   1. Initial program 46.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 45.3%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 45.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 45.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 45.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 51.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 51.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 51.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 51.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 51.5%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 71.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 81.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 81.8%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 81.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around 0 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   10. Step-by-step derivation

       1. *-commutative 71.6%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

       2. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

       3. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       4. associate-*r* 75.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       5. times-frac 94.3%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Simplified 94.3%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 7.99999999999999949e-49 < t

   1. Initial program 76.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 76.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. pow3 76.6%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. cbrt-prod 76.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\sin k}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. div-inv 76.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. cbrt-prod 77.7%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. unpow3 77.7%

         \[\leadsto \frac{2}{{\left(\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. add-cbrt-cube 84.3%

         \[\leadsto \frac{2}{{\left(\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. pow2 84.3%

         \[\leadsto \frac{2}{{\left(\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      9. pow-flip 84.3%

         \[\leadsto \frac{2}{{\left(\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      10. metadata-eval 84.3%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. Applied egg-rr 84.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]

Alternative 6: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt[3]{\sin k}\\ t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{t_2} \cdot t_1\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{{t_1}^{3}}}{t_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (cbrt (sin k)))) (t_2 (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
   (if (<= t -8.5e+43)
     (* (/ 2.0 (pow (* (cbrt t_2) t_1) 3.0)) (* l l))
     (if (<= t 8.5e-72)
       (/ 2.0 (* (* (/ (* t k) l) (/ k l)) (* (tan k) (sin k))))
       (* l (* l (/ (/ 2.0 (pow t_1 3.0)) t_2)))))))

C

double code(double t, double l, double k) {
	double t_1 = t * cbrt(sin(k));
	double t_2 = tan(k) * (2.0 + pow((k / t), 2.0));
	double tmp;
	if (t <= -8.5e+43) {
		tmp = (2.0 / pow((cbrt(t_2) * t_1), 3.0)) * (l * l);
	} else if (t <= 8.5e-72) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	} else {
		tmp = l * (l * ((2.0 / pow(t_1, 3.0)) / t_2));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t * Math.cbrt(Math.sin(k));
	double t_2 = Math.tan(k) * (2.0 + Math.pow((k / t), 2.0));
	double tmp;
	if (t <= -8.5e+43) {
		tmp = (2.0 / Math.pow((Math.cbrt(t_2) * t_1), 3.0)) * (l * l);
	} else if (t <= 8.5e-72) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = l * (l * ((2.0 / Math.pow(t_1, 3.0)) / t_2));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t * cbrt(sin(k)))
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (t <= -8.5e+43)
		tmp = Float64(Float64(2.0 / (Float64(cbrt(t_2) * t_1) ^ 3.0)) * Float64(l * l));
	elseif (t <= 8.5e-72)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / (t_1 ^ 3.0)) / t_2)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.5e+43], N[(N[(2.0 / N[Power[N[(N[Power[t$95$2, 1/3], $MachinePrecision] * t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e-72], N[(2.0 / N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[(2.0 / N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.5e43

   1. Initial program 58.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 58.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 58.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 59.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 59.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 59.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 59.6%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 80.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   if -8.5e43 < t < 8.50000000000000008e-72

   1. Initial program 45.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 45.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 44.7%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 44.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 44.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 44.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 44.6%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 44.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 44.6%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 44.6%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 43.8%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 43.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 43.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 51.0%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 51.0%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 51.0%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 51.0%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 51.0%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 71.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 81.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 81.4%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 81.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around 0 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   10. Step-by-step derivation

       1. *-commutative 71.6%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

       2. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

       3. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       4. associate-*r* 75.2%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       5. times-frac 94.2%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Simplified 94.2%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 8.50000000000000008e-72 < t

   1. Initial program 76.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 76.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 77.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 74.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 73.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 73.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 73.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 73.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 73.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 73.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 73.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 73.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 72.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}\right) \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}\right)}} \]

      2. pow3 72.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}\right)}^{3}}} \]

      3. cbrt-prod 72.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\color{blue}{\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt[3]{{t}^{3} \cdot \sin k}\right)}}^{3}} \]

      4. cbrt-prod 72.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\sin k}\right)}\right)}^{3}} \]

      5. unpow3 72.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      6. add-cbrt-cube 77.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\color{blue}{t} \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

   5. Applied egg-rr 77.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{{\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. pow1 77.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\tan k \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}\right)}^{1}}} \]

      2. *-commutative 77.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\tan k \cdot {\color{blue}{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}}^{3}\right)}^{1}} \]

   7. Applied egg-rr 77.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\tan k \cdot {\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}^{3}\right)}^{1}}} \]
   8. Step-by-step derivation

      1. unpow1 77.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot {\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}^{3}}} \]

      2. *-commutative 77.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}^{3} \cdot \tan k}} \]

      3. cube-prod 77.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}^{3}\right)} \cdot \tan k} \]

      4. rem-cube-cbrt 77.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right) \cdot \tan k} \]

      5. associate-*l* 77.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

   9. Simplified 77.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 77.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}\right)\right)} \]

       2. expm1-udef 72.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}\right)} - 1} \]

       3. associate-*l* 74.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}\right)}\right)} - 1 \]

       4. *-commutative 74.3%

          \[\leadsto e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}\right)\right)} - 1 \]

   11. Applied egg-rr 74.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 80.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\right)\right)} \]

       2. expm1-log1p 81.6%

          \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

       3. associate-/r* 81.6%

          \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\frac{\frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\right) \]

   13. Simplified 81.6%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{\frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 7: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{+45}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 \cdot k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.08e+45)
   (* (* l l) (/ 2.0 (pow (* (* t (cbrt (sin k))) (cbrt (* 2.0 k))) 3.0)))
   (if (<= t 8e-49)
     (/ 2.0 (* (* (/ (* t k) l) (/ k l)) (* (tan k) (sin k))))
     (/ 2.0 (* (/ (* k (pow t 3.0)) (* l l)) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.08e+45) {
		tmp = (l * l) * (2.0 / pow(((t * cbrt(sin(k))) * cbrt((2.0 * k))), 3.0));
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	} else {
		tmp = 2.0 / (((k * pow(t, 3.0)) / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.08e+45) {
		tmp = (l * l) * (2.0 / Math.pow(((t * Math.cbrt(Math.sin(k))) * Math.cbrt((2.0 * k))), 3.0));
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = 2.0 / (((k * Math.pow(t, 3.0)) / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.08e+45)
		tmp = Float64(Float64(l * l) * Float64(2.0 / (Float64(Float64(t * cbrt(sin(k))) * cbrt(Float64(2.0 * k))) ^ 3.0)));
	elseif (t <= 8e-49)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * (t ^ 3.0)) / Float64(l * l)) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.08e+45], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-49], N[(2.0 / N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.08e45

   1. Initial program 58.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 58.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 58.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 59.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 59.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 59.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 59.6%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 80.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]

   6. Taylor expanded in k around 0 75.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\color{blue}{2 \cdot k}} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]
   7. Step-by-step derivation

      1. *-commutative 75.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\color{blue}{k \cdot 2}} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

   8. Simplified 75.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\color{blue}{k \cdot 2}} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

   if -1.08e45 < t < 7.99999999999999949e-49

   1. Initial program 46.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 45.3%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 45.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 45.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 45.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 51.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 51.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 51.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 51.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 51.5%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 71.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 81.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 81.8%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 81.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around 0 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   10. Step-by-step derivation

       1. *-commutative 71.6%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

       2. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

       3. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       4. associate-*r* 75.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       5. times-frac 94.3%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Simplified 94.3%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 7.99999999999999949e-49 < t

   1. Initial program 76.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 70.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   5. Taylor expanded in k around 0 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
   6. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \frac{2}{\frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

   7. Simplified 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{+45}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 \cdot k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 8: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -7.5e+45)
   (/ 2.0 (* (* 2.0 k) (* (sin k) (pow (* t (cbrt (pow l -2.0))) 3.0))))
   (if (<= t 8e-49)
     (/ 2.0 (* (* (/ (* t k) l) (/ k l)) (* (tan k) (sin k))))
     (/ 2.0 (* (/ (* k (pow t 3.0)) (* l l)) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -7.5e+45) {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * pow((t * cbrt(pow(l, -2.0))), 3.0)));
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	} else {
		tmp = 2.0 / (((k * pow(t, 3.0)) / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -7.5e+45) {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * Math.pow((t * Math.cbrt(Math.pow(l, -2.0))), 3.0)));
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = 2.0 / (((k * Math.pow(t, 3.0)) / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -7.5e+45)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * (Float64(t * cbrt((l ^ -2.0))) ^ 3.0))));
	elseif (t <= 8e-49)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * (t ^ 3.0)) / Float64(l * l)) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -7.5e+45], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-49], N[(2.0 / N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.50000000000000058e45

   1. Initial program 58.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 58.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 58.4%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 53.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 47.8%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 47.8%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 47.8%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 47.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 47.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 64.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 64.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 64.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 64.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 68.1%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}} \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

   if -7.50000000000000058e45 < t < 7.99999999999999949e-49

   1. Initial program 46.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 45.3%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 45.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 45.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 45.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 51.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 51.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 51.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 51.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 51.5%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 71.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 81.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 81.8%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 81.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around 0 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   10. Step-by-step derivation

       1. *-commutative 71.6%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

       2. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

       3. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       4. associate-*r* 75.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       5. times-frac 94.3%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Simplified 94.3%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 7.99999999999999949e-49 < t

   1. Initial program 76.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 70.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   5. Taylor expanded in k around 0 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
   6. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \frac{2}{\frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

   7. Simplified 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 9: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+45}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -2.4e+45)
   (* (* l l) (/ 2.0 (* (pow (* t (cbrt (sin k))) 3.0) (* 2.0 k))))
   (if (<= t 8e-49)
     (/ 2.0 (* (* (/ (* t k) l) (/ k l)) (* (tan k) (sin k))))
     (/ 2.0 (* (/ (* k (pow t 3.0)) (* l l)) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.4e+45) {
		tmp = (l * l) * (2.0 / (pow((t * cbrt(sin(k))), 3.0) * (2.0 * k)));
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	} else {
		tmp = 2.0 / (((k * pow(t, 3.0)) / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.4e+45) {
		tmp = (l * l) * (2.0 / (Math.pow((t * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k)));
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = 2.0 / (((k * Math.pow(t, 3.0)) / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -2.4e+45)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((Float64(t * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k))));
	elseif (t <= 8e-49)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * (t ^ 3.0)) / Float64(l * l)) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -2.4e+45], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-49], N[(2.0 / N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.39999999999999989e45

   1. Initial program 58.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 58.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 58.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 59.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 59.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 59.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 59.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 59.6%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}\right) \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}\right)}} \]

      2. pow3 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}\right)}^{3}}} \]

      3. cbrt-prod 59.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\color{blue}{\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt[3]{{t}^{3} \cdot \sin k}\right)}}^{3}} \]

      4. cbrt-prod 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\sin k}\right)}\right)}^{3}} \]

      5. unpow3 59.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

      6. add-cbrt-cube 72.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\color{blue}{t} \cdot \sqrt[3]{\sin k}\right)\right)}^{3}} \]

   5. Applied egg-rr 72.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{{\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. pow1 72.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\tan k \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}\right)}^{1}}} \]

      2. *-commutative 72.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\tan k \cdot {\color{blue}{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}}^{3}\right)}^{1}} \]

   7. Applied egg-rr 72.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\tan k \cdot {\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}^{3}\right)}^{1}}} \]
   8. Step-by-step derivation

      1. unpow1 72.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot {\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}^{3}}} \]

      2. *-commutative 72.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}^{3} \cdot \tan k}} \]

      3. cube-prod 72.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}^{3}\right)} \cdot \tan k} \]

      4. rem-cube-cbrt 72.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right) \cdot \tan k} \]

      5. associate-*l* 72.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

   9. Simplified 72.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

   10. Taylor expanded in k around 0 67.0%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 67.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   12. Simplified 67.0%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   if -2.39999999999999989e45 < t < 7.99999999999999949e-49

   1. Initial program 46.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 45.3%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 45.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 45.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 45.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 51.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 51.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 51.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 51.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 51.5%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 71.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 81.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 81.8%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 81.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around 0 71.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   10. Step-by-step derivation

       1. *-commutative 71.6%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

       2. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

       3. unpow2 71.6%

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       4. associate-*r* 75.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       5. times-frac 94.3%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Simplified 94.3%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 7.99999999999999949e-49 < t

   1. Initial program 76.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 70.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   5. Taylor expanded in k around 0 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
   6. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \frac{2}{\frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

   7. Simplified 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+45}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 10: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot {t}^{3}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+167}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot t_1}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\frac{k}{\ell} \cdot \left(t \cdot k\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (pow t 3.0))))
   (if (<= t -6e+167)
     (* l (/ l (* k t_1)))
     (if (<= t 8e-49)
       (/ 2.0 (* (sin k) (* (tan k) (/ (* (/ k l) (* t k)) l))))
       (/ 2.0 (* (/ t_1 (* l l)) (* 2.0 k)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * pow(t, 3.0);
	double tmp;
	if (t <= -6e+167) {
		tmp = l * (l / (k * t_1));
	} else if (t <= 8e-49) {
		tmp = 2.0 / (sin(k) * (tan(k) * (((k / l) * (t * k)) / l)));
	} else {
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (t ** 3.0d0)
    if (t <= (-6d+167)) then
        tmp = l * (l / (k * t_1))
    else if (t <= 8d-49) then
        tmp = 2.0d0 / (sin(k) * (tan(k) * (((k / l) * (t * k)) / l)))
    else
        tmp = 2.0d0 / ((t_1 / (l * l)) * (2.0d0 * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.pow(t, 3.0);
	double tmp;
	if (t <= -6e+167) {
		tmp = l * (l / (k * t_1));
	} else if (t <= 8e-49) {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (((k / l) * (t * k)) / l)));
	} else {
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.pow(t, 3.0)
	tmp = 0
	if t <= -6e+167:
		tmp = l * (l / (k * t_1))
	elif t <= 8e-49:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (((k / l) * (t * k)) / l)))
	else:
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * (t ^ 3.0))
	tmp = 0.0
	if (t <= -6e+167)
		tmp = Float64(l * Float64(l / Float64(k * t_1)));
	elseif (t <= 8e-49)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(Float64(Float64(k / l) * Float64(t * k)) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 / Float64(l * l)) * Float64(2.0 * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (t ^ 3.0);
	tmp = 0.0;
	if (t <= -6e+167)
		tmp = l * (l / (k * t_1));
	elseif (t <= 8e-49)
		tmp = 2.0 / (sin(k) * (tan(k) * (((k / l) * (t * k)) / l)));
	else
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6e+167], N[(l * N[(l / N[(k * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-49], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(k / l), $MachinePrecision] * N[(t * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.00000000000000023e167

   1. Initial program 72.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 72.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 72.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 72.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 72.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 72.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 72.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 85.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. pow1 85.2%

         \[\leadsto \color{blue}{{\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)}^{1}} \]

      2. associate-*l* 85.7%

         \[\leadsto {\color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}}^{1} \]

   7. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}^{1}} \]

   8. Taylor expanded in k around 0 53.1%

      \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}}\right)}^{1} \]
   9. Step-by-step derivation

      1. unpow2 53.1%

         \[\leadsto {\left(\ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}\right)}^{1} \]

      2. associate-*l* 72.7%

         \[\leadsto {\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{1} \]

   10. Simplified 72.7%

       \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{1} \]

   if -6.00000000000000023e167 < t < 7.99999999999999949e-49

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 45.2%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 45.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 45.1%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 45.1%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 44.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 55.6%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 55.6%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 55.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 55.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 55.7%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 67.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 67.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 67.9%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 76.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 76.2%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 76.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 65.4%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 34.1%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. associate-/l* 36.2%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{k}{\frac{\ell}{k}}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   10. Applied egg-rr 36.2%

       \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   11. Step-by-step derivation

       1. expm1-def 69.7%

          \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

       2. expm1-log1p 80.7%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

       3. *-commutative 80.7%

          \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)}} \]

       4. associate-*l* 81.4%

          \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)\right)}} \]

       5. associate-*l/ 81.9%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\right)} \]

       6. *-commutative 81.9%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{\frac{k}{\frac{\ell}{k}} \cdot t}}{\ell}\right)} \]

       7. associate-/r/ 81.8%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot t}{\ell}\right)} \]

       8. associate-*l* 86.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{\frac{k}{\ell} \cdot \left(k \cdot t\right)}}{\ell}\right)} \]

   12. Simplified 86.0%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{\frac{k}{\ell} \cdot \left(k \cdot t\right)}{\ell}\right)}} \]

   if 7.99999999999999949e-49 < t

   1. Initial program 76.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 70.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   5. Taylor expanded in k around 0 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
   6. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \frac{2}{\frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

   7. Simplified 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+167}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\frac{k}{\ell} \cdot \left(t \cdot k\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 11: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot {t}^{3}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+167}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot t_1}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (pow t 3.0))))
   (if (<= t -4.8e+167)
     (* l (/ l (* k t_1)))
     (if (<= t 8e-49)
       (/ 2.0 (* (* (/ (* t k) l) (/ k l)) (* (tan k) (sin k))))
       (/ 2.0 (* (/ t_1 (* l l)) (* 2.0 k)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * pow(t, 3.0);
	double tmp;
	if (t <= -4.8e+167) {
		tmp = l * (l / (k * t_1));
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	} else {
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (t ** 3.0d0)
    if (t <= (-4.8d+167)) then
        tmp = l * (l / (k * t_1))
    else if (t <= 8d-49) then
        tmp = 2.0d0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)))
    else
        tmp = 2.0d0 / ((t_1 / (l * l)) * (2.0d0 * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.pow(t, 3.0);
	double tmp;
	if (t <= -4.8e+167) {
		tmp = l * (l / (k * t_1));
	} else if (t <= 8e-49) {
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.pow(t, 3.0)
	tmp = 0
	if t <= -4.8e+167:
		tmp = l * (l / (k * t_1))
	elif t <= 8e-49:
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (math.tan(k) * math.sin(k)))
	else:
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * (t ^ 3.0))
	tmp = 0.0
	if (t <= -4.8e+167)
		tmp = Float64(l * Float64(l / Float64(k * t_1)));
	elseif (t <= 8e-49)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 / Float64(l * l)) * Float64(2.0 * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (t ^ 3.0);
	tmp = 0.0;
	if (t <= -4.8e+167)
		tmp = l * (l / (k * t_1));
	elseif (t <= 8e-49)
		tmp = 2.0 / ((((t * k) / l) * (k / l)) * (tan(k) * sin(k)));
	else
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.8e+167], N[(l * N[(l / N[(k * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-49], N[(2.0 / N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.79999999999999998e167

   1. Initial program 72.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 72.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 72.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 72.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 72.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 72.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 72.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 85.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. pow1 85.2%

         \[\leadsto \color{blue}{{\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)}^{1}} \]

      2. associate-*l* 85.7%

         \[\leadsto {\color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}}^{1} \]

   7. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}^{1}} \]

   8. Taylor expanded in k around 0 53.1%

      \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}}\right)}^{1} \]
   9. Step-by-step derivation

      1. unpow2 53.1%

         \[\leadsto {\left(\ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}\right)}^{1} \]

      2. associate-*l* 72.7%

         \[\leadsto {\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{1} \]

   10. Simplified 72.7%

       \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{1} \]

   if -4.79999999999999998e167 < t < 7.99999999999999949e-49

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 45.2%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 45.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 45.1%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 45.1%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 44.5%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 44.4%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 55.6%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 55.6%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 55.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 55.7%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 55.7%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 67.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 67.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 67.9%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 76.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 76.2%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 76.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around 0 67.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   10. Step-by-step derivation

       1. *-commutative 67.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

       2. unpow2 67.9%

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

       3. unpow2 67.9%

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       4. associate-*r* 70.6%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

       5. times-frac 87.1%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Simplified 87.1%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 7.99999999999999949e-49 < t

   1. Initial program 76.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 70.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   5. Taylor expanded in k around 0 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
   6. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \frac{2}{\frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

   7. Simplified 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+167}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 12: 64.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{{k}^{4}}\\ t_2 := k \cdot {t}^{3}\\ t_3 := k \cdot t_2\\ \mathbf{if}\;t \leq -2.06 \cdot 10^{+167}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t_3}\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot t_1}{t}\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-104}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{t_3}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-51}:\\ \;\;\;\;t_1 \cdot \frac{2}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (pow k 4.0))) (t_2 (* k (pow t 3.0))) (t_3 (* k t_2)))
   (if (<= t -2.06e+167)
     (* l (/ l t_3))
     (if (<= t -1.65e+94)
       (* 2.0 (/ (* l t_1) t))
       (if (<= t -1.26e-104)
         (/ (* (* l l) (cos k)) t_3)
         (if (<= t 1.15e-51)
           (* t_1 (/ 2.0 (/ t l)))
           (/ 2.0 (* (/ t_2 (* l l)) (* 2.0 k)))))))))

C

double code(double t, double l, double k) {
	double t_1 = l / pow(k, 4.0);
	double t_2 = k * pow(t, 3.0);
	double t_3 = k * t_2;
	double tmp;
	if (t <= -2.06e+167) {
		tmp = l * (l / t_3);
	} else if (t <= -1.65e+94) {
		tmp = 2.0 * ((l * t_1) / t);
	} else if (t <= -1.26e-104) {
		tmp = ((l * l) * cos(k)) / t_3;
	} else if (t <= 1.15e-51) {
		tmp = t_1 * (2.0 / (t / l));
	} else {
		tmp = 2.0 / ((t_2 / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = l / (k ** 4.0d0)
    t_2 = k * (t ** 3.0d0)
    t_3 = k * t_2
    if (t <= (-2.06d+167)) then
        tmp = l * (l / t_3)
    else if (t <= (-1.65d+94)) then
        tmp = 2.0d0 * ((l * t_1) / t)
    else if (t <= (-1.26d-104)) then
        tmp = ((l * l) * cos(k)) / t_3
    else if (t <= 1.15d-51) then
        tmp = t_1 * (2.0d0 / (t / l))
    else
        tmp = 2.0d0 / ((t_2 / (l * l)) * (2.0d0 * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.pow(k, 4.0);
	double t_2 = k * Math.pow(t, 3.0);
	double t_3 = k * t_2;
	double tmp;
	if (t <= -2.06e+167) {
		tmp = l * (l / t_3);
	} else if (t <= -1.65e+94) {
		tmp = 2.0 * ((l * t_1) / t);
	} else if (t <= -1.26e-104) {
		tmp = ((l * l) * Math.cos(k)) / t_3;
	} else if (t <= 1.15e-51) {
		tmp = t_1 * (2.0 / (t / l));
	} else {
		tmp = 2.0 / ((t_2 / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.pow(k, 4.0)
	t_2 = k * math.pow(t, 3.0)
	t_3 = k * t_2
	tmp = 0
	if t <= -2.06e+167:
		tmp = l * (l / t_3)
	elif t <= -1.65e+94:
		tmp = 2.0 * ((l * t_1) / t)
	elif t <= -1.26e-104:
		tmp = ((l * l) * math.cos(k)) / t_3
	elif t <= 1.15e-51:
		tmp = t_1 * (2.0 / (t / l))
	else:
		tmp = 2.0 / ((t_2 / (l * l)) * (2.0 * k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / (k ^ 4.0))
	t_2 = Float64(k * (t ^ 3.0))
	t_3 = Float64(k * t_2)
	tmp = 0.0
	if (t <= -2.06e+167)
		tmp = Float64(l * Float64(l / t_3));
	elseif (t <= -1.65e+94)
		tmp = Float64(2.0 * Float64(Float64(l * t_1) / t));
	elseif (t <= -1.26e-104)
		tmp = Float64(Float64(Float64(l * l) * cos(k)) / t_3);
	elseif (t <= 1.15e-51)
		tmp = Float64(t_1 * Float64(2.0 / Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 / Float64(l * l)) * Float64(2.0 * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / (k ^ 4.0);
	t_2 = k * (t ^ 3.0);
	t_3 = k * t_2;
	tmp = 0.0;
	if (t <= -2.06e+167)
		tmp = l * (l / t_3);
	elseif (t <= -1.65e+94)
		tmp = 2.0 * ((l * t_1) / t);
	elseif (t <= -1.26e-104)
		tmp = ((l * l) * cos(k)) / t_3;
	elseif (t <= 1.15e-51)
		tmp = t_1 * (2.0 / (t / l));
	else
		tmp = 2.0 / ((t_2 / (l * l)) * (2.0 * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(k * t$95$2), $MachinePrecision]}, If[LessEqual[t, -2.06e+167], N[(l * N[(l / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.65e+94], N[(2.0 * N[(N[(l * t$95$1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.26e-104], N[(N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t, 1.15e-51], N[(t$95$1 * N[(2.0 / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -2.06e167

   1. Initial program 72.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 72.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 72.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 72.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 72.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 72.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 72.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 85.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. pow1 85.2%

         \[\leadsto \color{blue}{{\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)}^{1}} \]

      2. associate-*l* 85.7%

         \[\leadsto {\color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}}^{1} \]

   7. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}^{1}} \]

   8. Taylor expanded in k around 0 53.1%

      \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}}\right)}^{1} \]
   9. Step-by-step derivation

      1. unpow2 53.1%

         \[\leadsto {\left(\ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}\right)}^{1} \]

      2. associate-*l* 72.7%

         \[\leadsto {\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{1} \]

   10. Simplified 72.7%

       \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{1} \]

   if -2.06e167 < t < -1.65e94

   1. Initial program 35.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 35.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 35.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 35.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 35.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 35.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 35.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 35.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 35.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 35.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 35.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 35.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 64.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 64.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 64.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 64.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 64.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 64.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 64.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

   7. Taylor expanded in k around 0 64.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 64.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 64.8%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 65.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 65.1%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 65.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   11. Applied egg-rr 65.2%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   if -1.65e94 < t < -1.26e-104

   1. Initial program 75.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 75.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 75.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 75.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 75.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 75.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 75.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 75.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 75.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 75.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in t around inf 51.8%

      \[\leadsto \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 51.8%

         \[\leadsto \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot {t}^{3}} \]

   6. Simplified 51.8%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2} \cdot {t}^{3}}} \]

   7. Taylor expanded in k around 0 61.2%

      \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 56.9%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. associate-*l* 58.9%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   9. Simplified 63.4%

      \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   if -1.26e-104 < t < 1.15000000000000001e-51

   1. Initial program 32.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 32.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 32.1%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 32.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 32.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 32.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 32.0%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 32.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 32.0%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      2. pow3 32.0%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      3. div-inv 30.9%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      4. cbrt-prod 30.8%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. unpow3 30.8%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. add-cbrt-cube 41.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      7. pow2 41.2%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      8. pow-flip 41.3%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      9. metadata-eval 41.3%

         \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)}^{3}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 41.3%

      \[\leadsto \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)}^{3}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Taylor expanded in k around inf 67.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. *-commutative 67.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 67.4%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 80.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 80.1%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Simplified 80.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in k around 0 57.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. unpow2 57.1%

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

       2. *-commutative 57.1%

          \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

       3. times-frac 68.4%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

       4. associate-/r/ 68.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{t}{\frac{\ell}{{k}^{4}}}}} \]

       5. associate-*r/ 68.3%

          \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\frac{t}{\frac{\ell}{{k}^{4}}}}} \]

       6. associate-/r/ 68.1%

          \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}} \]

       7. times-frac 68.4%

          \[\leadsto \color{blue}{\frac{2}{\frac{t}{\ell}} \cdot \frac{\ell}{{k}^{4}}} \]

   11. Simplified 68.4%

       \[\leadsto \color{blue}{\frac{2}{\frac{t}{\ell}} \cdot \frac{\ell}{{k}^{4}}} \]

   if 1.15000000000000001e-51 < t

   1. Initial program 76.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 70.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   5. Taylor expanded in k around 0 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
   6. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \frac{2}{\frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

   7. Simplified 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.06 \cdot 10^{+167}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-104}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{\ell}{{k}^{4}} \cdot \frac{2}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 13: 67.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.3e-15)
   (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* 2.0 k)))
   (* (* l l) (/ 2.0 (* (tan k) (* (sin k) (* k (* t k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e-15) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	} else {
		tmp = (l * l) * (2.0 / (tan(k) * (sin(k) * (k * (t * k)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d-15) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (sin(k) / l)) * (2.0d0 * k))
    else
        tmp = (l * l) * (2.0d0 / (tan(k) * (sin(k) * (k * (t * k)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e-15) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (2.0 * k));
	} else {
		tmp = (l * l) * (2.0 / (Math.tan(k) * (Math.sin(k) * (k * (t * k)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.3e-15:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (2.0 * k))
	else:
		tmp = (l * l) * (2.0 / (math.tan(k) * (math.sin(k) * (k * (t * k)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.3e-15)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(2.0 * k)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(k * Float64(t * k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.3e-15)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	else
		tmp = (l * l) * (2.0 / (tan(k) * (sin(k) * (k * (t * k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.3e-15], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.30000000000000002e-15

   1. Initial program 58.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 58.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 58.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 59.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   5. Taylor expanded in t around 0 59.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
   6. Step-by-step derivation

      1. *-commutative 59.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot \sin k}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)} \]

      2. unpow2 59.6%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

      3. times-frac 64.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)} \cdot \left(2 \cdot k\right)} \]

   7. Simplified 64.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)} \cdot \left(2 \cdot k\right)} \]

   if 1.30000000000000002e-15 < k

   1. Initial program 55.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 55.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 55.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 55.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 55.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 55.4%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 55.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 55.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 55.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 55.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 55.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 55.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 70.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 70.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 70.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 70.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 70.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 73.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 73.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]

Alternative 14: 62.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(t \cdot {k}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6e+35)
   (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* 2.0 k)))
   (* (* l l) (/ 2.0 (* (tan k) (* t (pow k 3.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e+35) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	} else {
		tmp = (l * l) * (2.0 / (tan(k) * (t * pow(k, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6d+35) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (sin(k) / l)) * (2.0d0 * k))
    else
        tmp = (l * l) * (2.0d0 / (tan(k) * (t * (k ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e+35) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (2.0 * k));
	} else {
		tmp = (l * l) * (2.0 / (Math.tan(k) * (t * Math.pow(k, 3.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6e+35:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (2.0 * k))
	else:
		tmp = (l * l) * (2.0 / (math.tan(k) * (t * math.pow(k, 3.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6e+35)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(2.0 * k)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(t * (k ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6e+35)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (2.0 * k));
	else
		tmp = (l * l) * (2.0 / (tan(k) * (t * (k ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6e+35], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(t * N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.99999999999999981e35

   1. Initial program 60.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 60.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 60.6%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 60.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 58.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   5. Taylor expanded in t around 0 58.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
   6. Step-by-step derivation

      1. *-commutative 58.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot \sin k}}{{\ell}^{2}} \cdot \left(2 \cdot k\right)} \]

      2. unpow2 58.6%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

      3. times-frac 63.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)} \cdot \left(2 \cdot k\right)} \]

   7. Simplified 63.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)} \cdot \left(2 \cdot k\right)} \]

   if 5.99999999999999981e35 < k

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 48.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 48.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 48.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 48.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 48.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 48.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 48.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 48.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 48.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 48.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 48.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 68.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 68.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 68.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 68.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 72.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 72.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

   7. Taylor expanded in k around 0 55.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{3} \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(t \cdot {k}^{3}\right)}\\ \end{array} \]

Alternative 15: 62.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot {t}^{3}\\ \mathbf{if}\;t \leq -1.12 \cdot 10^{+167}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot t_1}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (pow t 3.0))))
   (if (<= t -1.12e+167)
     (/ (* l l) (* k t_1))
     (if (<= t 2.6e-50)
       (* 2.0 (/ (* l (/ l (pow k 4.0))) t))
       (/ 2.0 (* (/ t_1 (* l l)) (* 2.0 k)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * pow(t, 3.0);
	double tmp;
	if (t <= -1.12e+167) {
		tmp = (l * l) / (k * t_1);
	} else if (t <= 2.6e-50) {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	} else {
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (t ** 3.0d0)
    if (t <= (-1.12d+167)) then
        tmp = (l * l) / (k * t_1)
    else if (t <= 2.6d-50) then
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    else
        tmp = 2.0d0 / ((t_1 / (l * l)) * (2.0d0 * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.pow(t, 3.0);
	double tmp;
	if (t <= -1.12e+167) {
		tmp = (l * l) / (k * t_1);
	} else if (t <= 2.6e-50) {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	} else {
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.pow(t, 3.0)
	tmp = 0
	if t <= -1.12e+167:
		tmp = (l * l) / (k * t_1)
	elif t <= 2.6e-50:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	else:
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * (t ^ 3.0))
	tmp = 0.0
	if (t <= -1.12e+167)
		tmp = Float64(Float64(l * l) / Float64(k * t_1));
	elseif (t <= 2.6e-50)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 / Float64(l * l)) * Float64(2.0 * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (t ^ 3.0);
	tmp = 0.0;
	if (t <= -1.12e+167)
		tmp = (l * l) / (k * t_1);
	elseif (t <= 2.6e-50)
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	else
		tmp = 2.0 / ((t_1 / (l * l)) * (2.0 * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.12e+167], N[(N[(l * l), $MachinePrecision] / N[(k * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.6e-50], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.11999999999999999e167

   1. Initial program 72.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 72.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 72.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 72.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 72.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 72.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 72.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 72.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in t around inf 52.6%

      \[\leadsto \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 52.6%

         \[\leadsto \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot {t}^{3}} \]

   6. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2} \cdot {t}^{3}}} \]

   7. Taylor expanded in k around 0 52.6%

      \[\leadsto \frac{\color{blue}{{\ell}^{2}}}{{\sin k}^{2} \cdot {t}^{3}} \]
   8. Step-by-step derivation

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot {t}^{3}} \]

   9. Simplified 52.6%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot {t}^{3}} \]

   10. Taylor expanded in k around 0 52.6%

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. unpow2 52.6%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       2. associate-*l* 72.2%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   12. Simplified 72.2%

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   if -1.11999999999999999e167 < t < 2.6000000000000001e-50

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 46.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 46.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 46.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 46.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 46.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 46.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 47.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 47.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 47.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 47.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 67.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 67.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 67.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 67.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 67.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 70.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 70.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

   7. Taylor expanded in k around 0 57.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 57.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 57.2%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 63.5%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 63.5%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 63.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   11. Applied egg-rr 63.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   if 2.6000000000000001e-50 < t

   1. Initial program 76.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 70.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   5. Taylor expanded in k around 0 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
   6. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \frac{2}{\frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

   7. Simplified 75.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+167}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 16: 62.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+167} \lor \neg \left(t \leq 8 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -4.6e+167) (not (<= t 8e-49)))
   (/ (* l l) (* k (* k (pow t 3.0))))
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.6e+167) || !(t <= 8e-49)) {
		tmp = (l * l) / (k * (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-4.6d+167)) .or. (.not. (t <= 8d-49))) then
        tmp = (l * l) / (k * (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.6e+167) || !(t <= 8e-49)) {
		tmp = (l * l) / (k * (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -4.6e+167) or not (t <= 8e-49):
		tmp = (l * l) / (k * (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -4.6e+167) || !(t <= 8e-49))
		tmp = Float64(Float64(l * l) / Float64(k * Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -4.6e+167) || ~((t <= 8e-49)))
		tmp = (l * l) / (k * (k * (t ^ 3.0)));
	else
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -4.6e+167], N[Not[LessEqual[t, 8e-49]], $MachinePrecision]], N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.59999999999999976e167 or 7.99999999999999949e-49 < t

   1. Initial program 75.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 75.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 76.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 73.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 72.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 72.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 72.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 72.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 72.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 72.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 72.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 72.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in t around inf 58.4%

      \[\leadsto \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.4%

         \[\leadsto \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2} \cdot {t}^{3}} \]

   6. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2} \cdot {t}^{3}}} \]

   7. Taylor expanded in k around 0 58.4%

      \[\leadsto \frac{\color{blue}{{\ell}^{2}}}{{\sin k}^{2} \cdot {t}^{3}} \]
   8. Step-by-step derivation

      1. unpow2 58.4%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot {t}^{3}} \]

   9. Simplified 58.4%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot {t}^{3}} \]

   10. Taylor expanded in k around 0 63.5%

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. unpow2 63.5%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       2. associate-*l* 72.0%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   12. Simplified 72.0%

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   if -4.59999999999999976e167 < t < 7.99999999999999949e-49

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 46.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 46.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 46.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 46.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 46.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 46.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 47.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 47.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 47.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 47.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 67.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 67.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 67.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 67.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 67.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 70.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 70.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

   7. Taylor expanded in k around 0 57.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 57.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 57.2%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 63.5%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 63.5%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 63.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   11. Applied egg-rr 63.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+167} \lor \neg \left(t \leq 8 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 17: 59.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-46}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.8e-46)
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))
   (* (* l (pow t -3.0)) (/ l (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.8e-46) {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	} else {
		tmp = (l * pow(t, -3.0)) * (l / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 3.8d-46) then
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    else
        tmp = (l * (t ** (-3.0d0))) * (l / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.8e-46) {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	} else {
		tmp = (l * Math.pow(t, -3.0)) * (l / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.8e-46:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	else:
		tmp = (l * math.pow(t, -3.0)) * (l / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.8e-46)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	else
		tmp = Float64(Float64(l * (t ^ -3.0)) * Float64(l / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.8e-46)
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	else
		tmp = (l * (t ^ -3.0)) * (l / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.8e-46], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.7999999999999997e-46

   1. Initial program 50.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 50.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 50.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 49.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 49.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 49.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 50.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 50.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 50.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 50.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 50.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 64.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 64.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 64.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 64.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 64.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 66.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 66.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

   7. Taylor expanded in k around 0 55.4%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 55.4%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 55.4%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 60.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 60.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 61.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   11. Applied egg-rr 61.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   if 3.7999999999999997e-46 < t

   1. Initial program 76.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 76.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 77.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 74.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 72.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 72.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 72.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 72.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 72.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 72.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 72.7%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 66.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 66.0%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 66.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 68.8%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 68.8%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 68.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 60.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-udef 55.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]

      3. div-inv 55.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      4. pow-flip 55.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      5. metadata-eval 55.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 55.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. expm1-def 60.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-log1p 68.8%

         \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 68.8%

       \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-46}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]

Alternative 18: 62.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+83}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.5e+83)
   (* l (/ l (* k (* k (pow t 3.0)))))
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.5e+83) {
		tmp = l * (l / (k * (k * pow(t, 3.0))));
	} else {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.5d+83) then
        tmp = l * (l / (k * (k * (t ** 3.0d0))))
    else
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.5e+83) {
		tmp = l * (l / (k * (k * Math.pow(t, 3.0))));
	} else {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6.5e+83:
		tmp = l * (l / (k * (k * math.pow(t, 3.0))))
	else:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.5e+83)
		tmp = Float64(l * Float64(l / Float64(k * Float64(k * (t ^ 3.0)))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.5e+83)
		tmp = l * (l / (k * (k * (t ^ 3.0))));
	else
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.5e+83], N[(l * N[(l / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.5000000000000003e83

   1. Initial program 60.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 60.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 60.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 58.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 58.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 58.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 58.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 59.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 59.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 59.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 59.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 59.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 59.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

      2. pow3 59.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 69.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. pow1 69.7%

         \[\leadsto \color{blue}{{\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)}^{1}} \]

      2. associate-*l* 75.6%

         \[\leadsto {\color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}}^{1} \]

   7. Applied egg-rr 75.6%

      \[\leadsto \color{blue}{{\left(\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)\right)}^{1}} \]

   8. Taylor expanded in k around 0 57.3%

      \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}}\right)}^{1} \]
   9. Step-by-step derivation

      1. unpow2 57.3%

         \[\leadsto {\left(\ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}\right)}^{1} \]

      2. associate-*l* 62.7%

         \[\leadsto {\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{1} \]

   10. Simplified 62.7%

       \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{1} \]

   if 6.5000000000000003e83 < k

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 47.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 47.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 47.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 47.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 47.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 47.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 47.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 47.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 47.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 47.8%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 47.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 66.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 66.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

      2. associate-*l* 66.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      3. *-commutative 66.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      4. unpow2 66.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      5. associate-*l* 71.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   6. Simplified 71.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

   7. Taylor expanded in k around 0 55.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 55.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 55.9%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 56.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 56.4%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 56.6%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   11. Applied egg-rr 56.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+83}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 19: 56.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l (pow k 4.0)) (/ l t))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / pow(k, 4.0)) * (l / t));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l / (k ** 4.0d0)) * (l / t))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / Math.pow(k, 4.0)) * (l / t));
}

Python

def code(t, l, k):
	return 2.0 * ((l / math.pow(k, 4.0)) * (l / t))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / (k ^ 4.0)) * (l / t));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 57.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 58.1%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 56.6%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 56.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 56.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 56.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 56.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 56.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 56.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 56.9%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 56.9%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 59.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. *-commutative 59.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

   2. associate-*l* 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

   3. *-commutative 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

   4. unpow2 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

   5. associate-*l* 60.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

6. Simplified 60.7%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

7. Taylor expanded in k around 0 52.2%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 52.2%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 52.2%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 55.8%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 55.8%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

10. Final simplification 55.8%

    \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right) \]

Alternative 20: 56.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (* l (/ l (pow k 4.0))) t)))

C

double code(double t, double l, double k) {
	return 2.0 * ((l * (l / pow(k, 4.0))) / t);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
}

Python

def code(t, l, k):
	return 2.0 * ((l * (l / math.pow(k, 4.0))) / t)

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 57.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 58.1%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 56.6%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 56.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 56.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 56.6%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 56.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 56.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 56.9%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 56.9%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 56.9%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 59.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. *-commutative 59.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}} \]

   2. associate-*l* 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

   3. *-commutative 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

   4. unpow2 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

   5. associate-*l* 60.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

6. Simplified 60.7%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]

7. Taylor expanded in k around 0 52.2%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 52.2%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 52.2%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 55.8%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 55.8%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
10. Step-by-step derivation

    1. associate-*l/ 55.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

11. Applied egg-rr 55.9%

    \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

12. Final simplification 55.9%

    \[\leadsto 2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t} \]

Reproduce

herbie shell --seed 2023200 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))